Previous: Summer77
Next: Spring78
Preliminary Exam - Fall 1977
Problem 1
Let
Find a real matrix
![$B$](img2-1.gif)
such that
![$B^{-1}AB$](img3-1.gif)
is diagonal.
Problem 3
Let
![$T$](img10-1.gif)
be an
![$n\times n$](img11-1.gif)
complex matrix. Show that
if and only if all the eigenvalues of
![$T$](img13-1.gif)
have absolute value
less than
![$1$](img14-1.gif)
.
Problem 4
Let
![$P$](img15-1.gif)
be a linear operator on a finite-dimensional vector space over
a finite field. Show that if
![$P$](img16-1.gif)
is invertible, then
![$P^n=I$](img17-1.gif)
for
some positive integer
![$n$](img18-1.gif)
.
Problem 6
Let
![% latex2html id marker 822
$u:\mbox{$\mathbb{R}^{2}$} \to \mbox{$\mathbb{R}^{}$}$](img29-1.gif)
be the function defined by
![$u(x,y)=x^3-3xy^2$](img30-1.gif)
. Show that
![$u$](img31-1.gif)
is harmonic and find
![% latex2html id marker 832
$v:\mbox{$\mathbb{R}^{2}$} \to \mbox{$\mathbb{R}^{}$}$](img32-1.gif)
such that the function
![% latex2html id marker 838
$f: \mbox{$\mathbb{C}\,^{}$} \to \mbox{$\mathbb{C}\,^{}$}$](img33-1.gif)
defined by
is analytic.
Problem 7
Evaluate
where
![$n$](img36-1.gif)
is a positive integer.
Problem 8
Find all solutions of the differential equation
subject to the condition
![$x(0)=1$](img38-1.gif)
and
![$x'(0)=0$](img39-1.gif)
.
Problem 9
Let
![% latex2html id marker 895
$f:[0,1] \to \mbox{$\mathbb{R}^{}$}$](img40-1.gif)
be continuously differentiable, with
![$f(0)=0$](img41-1.gif)
.
Prove that
Problem 10
Let
![% latex2html id marker 928
$f_n:\mbox{$\mathbb{R}^{}$} \to \mbox{$\mathbb{R}^{}$}$](img43-1.gif)
be differentiable for each
![$n = 1, 2, \ldots$](img44-1.gif)
with
![$\vert f_n'(x)\vert\leq 1$](img45-1.gif)
for all n and x. Assume
for all x. Prove that
![% latex2html id marker 938
$g:\mbox{$\mathbb{R}^{}$}\to\mbox{$\mathbb{R}^{}$}$](img47-1.gif)
is continuous.
Problem 11
Show that the differential equation
![$x'=3x^2$](img48-1.gif)
has no solution such
that
![$x(0)=1$](img49-1.gif)
and
![$x(t)$](img50-1.gif)
is defined for all real numbers t.
Problem 12
Let
![% latex2html id marker 970
$X \subset \mbox{$\mathbb{R}^{}$}$](img51-1.gif)
be a nonempty connected set of real numbers. If
every element of
![$X$](img52-1.gif)
is rational, prove
![$X$](img53-1.gif)
has only one element.
Problem 13
Consider the following four types of transformations:
Here,
![$z$](img56-1.gif)
is a variable complex number and the other letters denote
constant complex numbers. Show that each transformation takes
circles to either circles or straight lines.
Problem 14
If
![$a$](img57-1.gif)
and
![$b$](img58-1.gif)
are complex numbers and
![$a\neq 0$](img59-1.gif)
, the set
![$a^b$](img60-1.gif)
consists of those complex numbers
![$c$](img61-1.gif)
having a logarithm of the form
![$b\alpha$](img62-1.gif)
, for some logarithm
![$\alpha$](img63-1.gif)
of
![$a$](img64-1.gif)
. (That is,
![$e^{b\alpha}=c$](img65-1.gif)
and
![$e^{\alpha}=a$](img66-1.gif)
for some complex number
![$\alpha$](img67-1.gif)
.)
Describe set
![$a^b$](img68-1.gif)
when
![$a=1$](img69-1.gif)
and
![$b=1/3+i$](img70-1.gif)
.
Problem 15
Let
![% latex2html id marker 1048
$f:\mbox{$\mathbb{R}^{n}$}\to\mbox{$\mathbb{R}^{}$}$](img71-1.gif)
have continuous partial derivatives and satisfy
for all
![$x=(x_1,\ldots,x_n)$](img73-1.gif)
,
![$j=1,\ldots,n$](img74-1.gif)
. Prove that
(where
![$\Vert u\Vert=\sqrt{u_1^2+\cdots+u_n^2}\,$](img76-1.gif)
).
Problem 16
Let
![$E$](img77-1.gif)
and
![$F$](img78-1.gif)
be vector spaces (not assumed to be finite-dimensional).
Let
![$S : E \to F$](img79-1.gif)
be a linear transformation.
- Prove
is a vector space.
- Show
has a kernel
if and only if
is injective (i.e.,
one-to-one).
- Assume
is injective; prove
is linear.
Problem 17
Let
![$G$](img86-1.gif)
be the set of 3
![$\times$](img87-1.gif)
3 real matrices with zeros below
the diagonal and ones on the diagonal.
- Prove
is a group under matrix multiplication.
- Determine the center of
.
Problem 18
Suppose the complex number
![$\alpha$](img90-1.gif)
is a root of a polynomial of
degree
![$n$](img91-1.gif)
with rational coefficients. Prove that
![$1/\alpha$](img92-1.gif)
is
also a root of a polynomial of degree
![$n$](img93-1.gif)
with rational coefficients.
Problem 19
Let
![$M$](img94-1.gif)
be a real 3
![$\times$](img95-1.gif)
3 matrix such that
![$M^3 = I$](img96-1.gif)
,
![$M \neq I$](img97-1.gif)
.
- What are the eigenvalues of
?
- Give an example of such a matrix.
Problem 20
Let
![% latex2html id marker 1159
$\mathbb{C}\,^{3}$](img99-1.gif)
denote the set of ordered triples of complex numbers.
Define a map
![% latex2html id marker 1165
$F:\mbox{$\mathbb{C}\,^{3}$} \to \mbox{$\mathbb{C}\,^{3}$}$](img100-1.gif)
by
Prove that
![$F$](img102-1.gif)
is onto but not one-to-one.
Previous: Summer77
Next: Spring78
Paulo Ney de Souza & Jorge-Nuno Silva
2000-08-10